The role of Dirac equations in the classical mechanics of the relativistic top

TL;DR

This paper derives a classical mechanics framework for a relativistic spinning particle using Dirac equations within a conformal geometric setting, proposing a new approach called 'Affine Quantum Mechanics' that could unify fundamental forces.

Contribution

It provides a rigorous derivation of Dirac's equation squared from classical mechanics using conformal geometry, introducing 'Affine Quantum Mechanics' as a novel theoretical framework.

Findings

Derivation of Dirac's equation squared from classical principles.

Linearization of the Hamilton-Jacobi equation via an ansatz interpreted as a quantum wave function.

Potential extension to general relativistic space-times.

Abstract

A rigorous \textit{ab initio} derivation of the (square of) Dirac's equation for a single particle with spin is presented. The general Hamilton-Jacobi equation for the particle expressed in terms of a background Weyl's conformal geometry is found to be linearized, exactly and in closed form, by an \textit{ansatz} solution that can be straightforwardly interpreted as the "quantum wave function" $\psi_4$ of the 4-spinor Dirac's equation. In particular, all quantum features of the model arise from a subtle interplay between the conformal curvature of the configuration space acting as a potential and Weyl's "pre-potential" closely related to $\psi_4$, which acts on the particle trajectory. The theory, carried out here by assuming a Minkowsky metric, can be easily extended to arbitrary space-time Riemann metric, e.g. the one adopted in the context of General Relativity. This novel theoretical scenario, referred to as "Affine Quantum Mechanics", appears to be of general application and is expected to open a promising perspective in the modern endeavor aimed at the unification of the natural forces with gravitation.